Satellite Selection for Multi-Constellation SBAS

Todd Walter, Juan Blanch

Stanford University

Victoria Kropp University FAF Munich

ABSTRACT The incorporation of multiple constellations into satellite based augmentation systems (SBAS) may lead to cases where there are more corrected satellites in view than a receiver has tracking channels. This paper addresses two related topics: identifying the most important satellites to track in order to provide availability; and identifying a recommended number of channels. Previously, the SBAS minimum operational performance standards (MOPS) specified a minimum number of channels required for the user receiver. It is possible to obtain significantly worse availability with this minimum number than with the all-in-view solution when employing a poor satellite selection algorithm. Alternatively, it is possible to achieve high availability with fewer than the minimum number of channels and a very good selection algorithm. This paper describes example selection methods that achieve high availability. It further describes a method to specify performance instead of a minimum hardware channel capacity. This form of specification allows for greater flexibility in receiver design. Manufacturers would be allowed to choose between more channels combined with a simpler algorithm versus fewer channels and a more sophisticated algorithm. INTRODUCTION The advent of multiple constellations provides the opportunity to eliminate geometry weakness as a source of SBAS unavailability. GPS occasionally has areas where there is an insufficient density of satellites to support all desired operations. This most often occurs when a primary slot satellite is out of service. However, adding one or more constellations easily compensates for

this geometric shortcoming. In fact, we may now experience the opposite problem of having more satellites than may be tracked by the receiver. There are many possible methods for selecting a set of satellites to use for the GPS position solution. Very often, elevation angle is used to rank satellites. A receiver may sort the satellites by their elevation angle and keep the k highest. While this choice is good from a tracking robustness point of view, it does not lead to the best availability. Ideally, when choosing from n total satellites in view, the user will be able to find k that produce protection level values that are below the required alert limits. In general, it is desirable to find an algorithm that minimizes the vertical protection level (VPL) and the horizontal protection level (HPL) [1]. A brute force search through all combinations yields the optimal set, but may be costly and impractical for when there are many possible satellite subsets. We examine and compare several methods that are more practical than the “optimal” brute force search. One such method is a “greedy” algorithm that removes the single least important satellite one at a time until only k satellites remain. An important consideration is that the optimal set of satellites depends on the specific protection level being minimized. The best sets will be different for SBAS VPL and SBAS HPL. Therefore, we need to define a balance when choosing between deselecting a satellite that least affects the VPL versus deselecting a satellite that least affects the HPL. Another factor is that the receiver is also capable of reverting to advanced receiver autonomous integrity monitoring (ARAIM) [2-4] when leaving the SBAS service area or in the event of an SBAS outage. The optimal satellite sets for ARAIM VPL and HPL are further different from the SBAS sets. Thus, there may be another desirable goal to find a satellite set that simultaneously allows the SBAS and ARAIM VPLs and HPLs to remain below their respective alert limits.

We then use these algorithms to evaluate the decrease in performance relative to the all-in-view protection levels. We perform this analysis for dual constellation conditions in order to examine sensitivity to satellite redundancy and geometric strength. Later, different constellation scenarios should be evaluated to determine the robustness of the techniques to initial geometric strength and total numbers of satellites. This paper addresses several important questions:

• How quickly the protection levels increase as the number of tracking channels is decreased?

• How should tracking requirements be specified? • If we specify a minimum number of channels,

what is the correct value? PRIOR SATELLITE SELECTION ALGORITHMS Specifying a large required number of tracking channels does not automatically assure good performance. There will likely always be cases where the receiver cannot track all satellites in view and has to choose which ones to track and which to ignore. A poor selection algorithm can lead to poor performance even when the number of satellites tracked is large. Conversely a relatively small number of satellites may lead to good performance if those satellites are well chosen. This section will describe some commonly understood methods for satellite selection. Probably the most common method is to use the elevation angle as a discriminator. The receiver may determine elevation angle given a rough position estimate and the satellite almanac files that describe the approximate satellite locations. The user does not need to track the satellites to estimate their elevation angle for the assumed location. The receiver will determine the elevation angle for every satellite for which it has almanac data. It can then eliminate from consideration all of those satellites whose elevation angle falls below some elevation mask (e.g. 5° as in today’s GPS aviation receivers). If the receiver has enough channels to track all of the remaining satellites then no further selection is required. However, if there are more satellites remaining than tracking channels, the receiver must choose a set of satellites to track (or equivalently, the complementary set of satellites to exclude). The “elevation” method sorts the satellites by elevation angle and keeps the k satellites with the largest values. If there are more satellites above the mask than there are tracking channels, the lowest elevation satellites are

excluded. The lowest elevation satellites typically also have the lowest received power and are the most vulnerable to loss due to aircraft banking. However, they are often quite important for good vertical geometry. Removing the lowest satellites can significantly increase the vertical dilution of precision (VDOP) and in turn VPL for SBAS and ARAIM. Further, the elevation method does not take into account satellite health or weighting factors. Higher elevation satellites may be unmonitored by SBAS or have large variances associated with their corrections. Simply looking at elevation angle discards this additional information. A better method would also make use of the health and weighting information that is broadcast from the SBAS satellites. This information should be used together with the satellite locations. Only satellites designated as healthy by the SBAS should be included in the n satellites to be considered for tracking. An “optimal” brute force method would look at all possible combinations of k out of n satellites to determine the best performance. This method is optimal in terms of returning the best possible outcome, but is distinctly non-optimal in terms of computational cost. If there were n healthy satellites above the mask, a receiver with k channels would have to evaluate Nopt geometries where Nopt is given by

Nopt =n!

n− k( )! k! (1)

If n = 30 and k = 24, then Nopt = 593,775 geometries to evaluate. As n becomes larger or k becomes smaller, the number of geometries to evaluate becomes even larger. Although it is possible to efficiently code this evaluation in terms of measurement downdates rather than as separate matrix inversions, this approach has significant computational cost. We have only used it for a few isolated geometries to compare the optimal result to the results from other methodologies. The “greedy” method is similar to the optimal in that it evaluates the performance of the subsets [5]. The key difference is that the greedy method removes one satellite at a time and then uses the resulting geometry with the corresponding satellite removed to evaluate the next iteration. For a case with 30 initial satellites, all 30 subsets containing 29 satellites are evaluated. Then the one with the best metric is used for the next step where 29 subsets each containing 28 satellites are evaluated. This continues until only the desired number of satellites remains. The number of subsets to be evaluated by this method, Ngreedy, is given by:

Ngreedy =12n n+1( )− k k +1( )"# $% (2)

For n = 30 and k = 24, then Ngreedy = 165 geometries to evaluate. This is certainly more work that the elevation angle method, but far less than the optimal. Ideally, we would like to find a method that has an even smaller computational cost. Other selection algorithms have been developed primarily to minimize the geometrical dilution of precision (GDOP) [6-8]. Most maximize the volume of a polyhedron defined by the satellite locations. However, such methods do not account for the SBAS weights and are therefore not as well suited for our application. PERFORMANCE OPTIMIZATION In this section we will quantitatively define how we evaluate performance and therefore how we rank one set of satellites as being better than another set. The desired property is to maximize availability for SBAS operations. SBAS provides different service levels with different horizontal and vertical alert limits. If the receiver knows the vertical alert limit VAL and the horizontal alert limit (HAL) [9], it could utilize a cost function designed to try to keep the VPL and the HPL below these thresholds. Such cost functions would be small while the protection levels are below their respective alert limits but would dramatically increase as the protection approach or exceed these thresholds. However, some classes of SBAS receiver merely output position estimates and protection levels. They do not know which service levels or alert limits are being targeted. Such receivers do not know how much margin they have against the thresholds. In the more demanding SBAS services, the VAL is smaller than the HAL. Also, the user almost always has a larger VPL than HPL. Therefore, it is typically much more important to minimize the VPL than it is to keep the HPL small. However, one should take both into account and try to prevent either one from exceeding their respective alert limits. We have therefore chosen to use the following metric for ranking geometries: !

!𝐻𝑃𝐿! + 𝑉𝑃𝐿! (3)

where

𝐻𝑃𝐿 = 6 !

!𝑐!,! + 𝑐!,! + !

!𝑐!,! − 𝑐!,! + 𝑐!,!! ,

𝑉𝑃𝐿 = 5.33 𝑐!,!,

(4)

and C = GT ⋅W ⋅G( )

−1 (5)

is the position estimate covariance matrix in the East-North-Up (ENU) frame, G is the geometry matrix (also in the ENU frame), and W is the weighting matrix [1]. This cost function represents a trade between the vertical horizontal protection levels. The factor of ¼ multiplying the HPL shifts priority to minimizing the VPL over minimizing the HPL. This factor is arbitrary and could easily be adjusted to shift the balance in one direction or the other. Indeed, the cost function itself was subjectively chosen. It was chosen in large part due to its simplicity. We had initially optimized only the VPL, but found that sometimes satellite sets were chosen that had large HPL values. We found that by including the horizontal terms as in (3) we prevented large growth in the HPL. There are likely other cost functions that would lead to superior availability, however we believe that (3) is reasonable first choice. MEASUREMENT DOWNDATE METHOD Because we are trying to optimize elements of the covariance matrix, we return to the approach of the greedy algorithm. It is trying to identify the subset with the smallest value for (3). Rather than performing n separate matrix inversions to find the n subset versions of C, we can obtain them through

C i( ) =C+Si ⋅Si

T

pi,i (6)

where C(i) is the position covariance matrix with the ith satellite removed, Si is the ith column of the S matrix S = GT ⋅W ⋅G( )

−1⋅GT ⋅W (7)

and P =W−W ⋅G ⋅ GT ⋅W ⋅G( )

−1⋅GT ⋅W (8)

Thus, starting from a single matrix inversion to obtain the all-in-view position estimate covariance matrix C, we can then find all the subset position estimate covariance matrices using much less costly matrix multiplications rather than inversions. While this downdate method points to a more efficient means to implement the greedy algorithm, we can see that it also points the way to an even more efficient algorithm. From (6) we can see that

cj, j( ) i( ) = cj, j +sj,i2

pi,i (9)

The last term in (9) represents the increase in the covariance matrix, along the jth axis, when removing the ith satellite. The smaller this term is, the less impact it has in increasing the corresponding covariance term. Therefore, if we calculate

14s1,i2 + s2,i

2( )+ s3,i2

pi,i (10)

and find the minimum value over all satellites, i, we will approach the cost function of (3). Most often we will have identified the satellite the greedy algorithm would choose to exclude at the first step. However, rather than following the greedy algorithm and calculating the covariance matrices for sub-subsets, we can simply sort the values in (10) from the all-in-view calculation and retain the satellites corresponding to the k largest values. We will call this the “downdate” method. We can see that it is much more efficient than the greedy method. Like the elevation angle method, we determine a set of values once for the all-in-view solution and then use the satellites with the k largest values. In the elevation method, the values are the elevation angles. In the downdate method, the values are given by (10). Although it requires more effort to determine these values than it does to determine the elevation angles, the downdate method is still very efficient compared to other alternatives. A similar method was recently proposed for GBAS [10], that uses s3,i to sort satellites. Although s3,i correlates well with VPL, the authors had to add further logic to ensure the minimum VPL was found. For SBAS, s3,i

2 / pi,i is proportional to ΔVPL2, so excluding the satellite with the minimum value corresponds to finding the one-out subset with the smallest VPL. In the next sections we will

compare the ability of the various selection routines to optimize performance. SIMULATION SETUP We used our Matlab algorithm availability simulation tool (MAAST) [11] to create simulated geometries and weights. In order to test the algorithms’ performance against a large number of potential satellites in view, we used a GPS almanac with 31 satellites (May 6, 2016), a Galileo almanac with 30 satellites, and the three active WAAS geostationary satellites. We simulated both the current single-frequency (SF) integrity algorithm performance and future dual-frequency (DF) algorithm performance. We evaluated performance for users spaced on a 2° x 2° grid and we used 300 evenly spaced time steps over one sidereal day. Users were constrained to be in a box between 15°N and 75°N, and between 175°W and 50°W. This set up was expected to create many different user scenarios, including ones where many satellites were in view, but with very different weights. The weights in particular are subject to variability. It is uncertain what values for will be obtained for the weighting terms by different SBAS providers, especially in a future DF environment. Thus, the absolute values of the protection levels are subject to change, but we believe that the relative percentage change due to removing satellites should be representative. The SF simulation created 158,788 valid position estimates with 23,768 of them having more than 24 usable satellites in view. The DF simulation created 188,200 valid position estimates with 26,709 having more than 24 usable satellites. Figure 1 shows histograms for the relative numbers in view for each case. The maximum

Figure 1. Relative occurrences of numbers of satellites in view for simulated users with valid position solutions

number in view for this constellation configuration was 31 satellites. The different simulations created a wide variety of user scenarios featuring different weighting and geometry conditions. We then applied the elevation, greedy, and downdate methods to simulate receivers that had differing values for the maximum number of satellites that could be tracked. EXAMPLE GEOMETRY Figure 2 shows a skyplot for an example geometry corresponding to the dual frequency simulation and for a user with 31 satellites in view. The numbers in the circles correspond to the PRN/SBAS slot numbers where values from 1 to 32 correspond to GPS, 75 to 111 to Galileo, and 120-158 to SBAS GEOs. The coloring indicates the sigma values used to create the weighting matrix. The SBAS GEOs, as is typical, have much higher sigmas and therefore much lower weighting. Figure 3 shows which satellites are excluded by the elevation, greedy, or downdate method assuming a maximum of 24 satellites. Those excluded by the elevation method are indicated by the blue top pie wedge, those excluded by the greedy method are indicated by the cyan bottom left pie wedge, and those excluded by the downdate method are indicated by the yellow bottom right pie wedge. Note that there is much better agreement between the greedy and downdate methods than there is with the elevation method. Both greedy and downdate agree that PRNs 12 and 92 are the least important satellites. They also both exclude 11, 93, 94, and 104, but not in the same order. Greedy also excludes 103 while downdate also removes 22. Both see relatively small increases in the VPL (3 cm for greedy and 2 cm for

downdate) and somewhat larger increases in HPL (70 cm for greedy and 98 cm for downdate). Both increases are much smaller than the increases seen by the elevation angle method (3.48 m in VPL and 1.23 m in HPL). Table 1 shows the HPLs and VPLs for four methods and for the maximum number of channels ranging from 31 down to 20. We also evaluated the optimal brute force method for this table. The downdate, greedy and optimal methods are all comparable, even when removing 11 out of 31 satellites. This is particularly impressive because the downdate method only calculates the S and P matrices one time, for the full all-in-view solution. These matrices are not reevaluated after each satellite removal, as is the case for the greedy and optimal methods. While these methods may not completely agree on the order in which to remove satellites, we find that there is little difference in performance. They are choosing between roughly equally important satellites, so the exact ranking is not critical. Contrast this to the elevation method, which is clearly removing satellites that otherwise to keep the VPL

Figure 2. Skyplot for example geometry showing satellite locations and confidence parameter.

Figure 3. Skyplot showing which satellites are excluded by either the elevation, greedy, or downdate method.

Table 1. SBAS VPL and HPL for differing numbers of channels and the different selection methods.

small. It is debatable which method truly performs better as it is not obvious how much more important it is to minimize VPL over HPL in this case. The last three methods all achieve VPLs below 10 m and HPLs below 6 m. Table 2 shows the order in which satellites are removed when excluding satellites by each of the different methods. When the number of channels is reduced by one, a single satellite is excluded from the prior set for the elevation, downdate, and greedy methods. This satellite will not be used for any cases with an even smaller number of channels. The optimal method, however, completely reevaluates the each possible set of satellites. Thus, sometimes satellites that were excluded for a particular number of channels are not excluded for a smaller number of channels. For example, the difference between the satellite set for the optimal method when going from 26 to 25 channels is to reintroduce PRN 11 and remove PRNs 22 & 103. In the next section we look at statistical performance for the full set of users and time steps. SIMULATION RESULTS Instantaneous availability is determined by whether the VPL and HPL are below their respective alert limits. The example geometry has an all-in-view VPL of 8.88 m and HPL of 4.34 m. These are well below the LPV-200 alert limits (VAL = 35 m & HAL = 40 m). They are even below the CAT-I autoland alert limits (VAL = 10 m & HAL = 40 m). The HPL could be significantly increased without crossing its threshold, however the VPL has substantially less margin. This is what motivated the factor of four dividing the horizontal terms in our cost function (3). Note that in the example geometry the elevation angle method does not support CAT-I autoland

with fewer than 27 channels, while the other methods support this mode down to at least 20 channels. Remember that the weighting values are subject to change as they depend on future dual-frequency algorithms. If the weights were made three times larger, the VPLs and HPLs would also all be three times larger. In that case, the all-in-view solution would still support LPV-200 (but not CAT-I autoland). In that scenario, the elevation angle method would not support LPV-200 with fewer than 25 channels and again the other methods would support it for down to at least 20 channels. Figure 4 shows the maximum observed percentage increase in the protection levels seen for the single frequency simulation. These values decrease as the number of channels increases. The elevation method has significantly larger values for the VPL, ranging from a ~5% increase at 30 channels, to more than a 100% increase for 20 channels. At 24 channels there was nearly a 50% increase. The downdate and greedy methods show dramatically smaller increases in VPL. They range from less than a 0.2% increase at 30 channels to less than 9% for 20 channels. These methods saw a ~2% maximum increase at 24 channels. The HPL increases for the three methods are much more similar, but the greedy method has the best performance. For 24 channels, the elevation and downdate methods see up to ~30% increase while the greedy method sees up to ~20%. A similar set of curves was obtained for the dual frequency simulation. Availability is typically specified as an average over time requiring more than 99% (or even higher) of all geometries at a single location to be instantaneously available. We can calculate the impact of having a limited number of channels has on observed availability,

Channels Elevation Downdate Greedy Optimal 30 3 12 12 12 29 91 92 92 92 28 101 104 94 94 27 138 93 11 11 26 18 22 104 104 25 83 94 103 (11), 22, 103 24 95 11 93 11, (22), 93

23 137 17 17 (11), 17, 22, 24, (103)

22 31 103 77 138 21 19 24 138 10, 103, (138) 20 85 10 137 138

Table 2. The order in which PRNs excluded by each method when decreasing the maximum number of channels. For the optimal method, a previously excluded satellite is sometimes returned (PRN in parentheses) and two others are excluded in its place.

Figure 4. The maximum observed percentage increases of VPL and HPL for different numbers of channels.

but it is harder to generalize the results. They will be very dependent on the assumed constellations and weights. They will also be very dependent on alert limits for the desired operation. Figure 4 shows the largest observed protection level increases, however if such large increases are only rarely observed, they may have little impact on average availability. However, if we evaluate constellations with an even greater number of satellites, the large increases in protection levels will be more common and will have a larger impact on average availability. Table 3 shows the percent decrease in CAT-I autoland coverage area for the dual-frequency simulation. The coverage region is the area in which a specified availability is met. We determined the coverage region for the all-in-view case corresponding to availabilities ranging from 95% to 100% and then compared them to the corresponding regions for different numbers of channels. There were no changes seen by any of the methods when 26 or more channels were considered. The elevation method saw some significant decreases when going below 24 channels. The downdate and greedy methods saw small decreases below 23 channels. The elevation method results indicate that the maximum increases indeed only affected relatively few geometries for our simulated scenarios. However, these results are very dependent on the assumption in the simulation scenario. A scenario with even more satellites or worse weights would see larger impacts at a higher number of channels. PERFORMANCE SPECIFICATION There is uncertainty surrounding the future number of satellites on orbit that will be corrected by SBAS. Therefore we advocate a MOPS requirement that will ensure high availability even if more satellites than anticipated are launched. The elevation method has the very undesirable property that with more satellites in view, the protection levels become potentially worse. This is because adding satellites at higher elevation will cause the receiver to discard lower elevation satellites,

raising its effective mask angle. A higher mask angle leads to larger VDOPs and VPLs. Instead we would like to encourage the use of a better selection method such as the downdate or greedy methods. These methods are very robust to differing numbers of satellites in view and only perform better as there are more satellites. However, we do not wish to mandate a particular algorithm as receiver manufacturers may have even better options available to them. Instead we propose to specify a reference set of geometries and weights. Each geometry would include the elevation and azimuth angles, to which constellation the satellite belongs, and the variances used to create the weighting matrix. We would also specify a maximum allowed VPL and HPL for each geometry. This information would be included as part of a Matlab tool that would allow a manufacturer to encode their selection algorithm and evaluate it its performance against each geometry. The specified algorithm is acceptable provided the tool confirms that the algorithm always returns protection levels below the thresholds. The thresholds would be set such that the downdate algorithm would pass the test, perhaps with some added margin. We still need to determine an appropriate number as well as which geometries to include. We envision that the tool could easily run hundreds if not thousand of simulated cases. We would include geometries that are both representative of potential future satellite configurations and that do not perform well with the elevation selection method. These scenarios need to be agreed upon by the wider SBAS community. COMPATIBILITY WITH ARAIM This paper has, so far, only dealt with satellite selection to optimize SBAS. However, DFMC SBAS receivers will also support ARAIM and will revert to this mode when out of SBAS coverage. Therefore, it is logical to want to optimize the SBAS and the ARAIM horizontal and vertical services. While the user may only need either SBAS or ARAIM service for any given operation, there is an advantage to having both available in case of a failure or an outage of the primary service. However, it is not always obvious what the best tradeoff between the two services is. A cost function that combines together the protection levels for both services would simultaneously limit the growth of each term, yet may fail to provide desired service through either. In contrast, a scheme that

Channels Elevation Downdate Greedy 25 0.11 0 0 24 1.26 0 0 23 5.78 0 0 22 13.24 0.06 0.06 21 29.19 0.09 0.09 20 51.79 0.29 0.25

Table 3. The observed percent decrease in CAT-I user coverage region for different numbers of channels.

optimizes either SBAS or ARAIM over the other, might be able to provide service through that one, at the expense of the other. ARAIM optimization is a little more difficult than for SBAS because the user will not necessarily know what confidence to place on a specific satellite until after it is already tracking it. The SBAS geostationary satellite broadcasts all of the confidence parameters for all of the GNSS satellites regardless of whether the user is tracking them or not. The SBAS user has full knowledge of the W and G matrices. In offline ARAIM [3], the URA/SISA value is only included in the ephemeris data broadcast from each satellite. The ARAIM user can only guess at the contribution to the W matrix before devoting a channel to track and gather the required data. Currently, GPS constellation broadcasts a URA value of 2.4 m more than 90% of the time [12], so this confidence value is not necessarily difficult to predict. However, it remains to be seen how predictable these values will be in the future with new constellations and new messaging capability on GPS. Nevertheless, we will assume that the URA/SISA values usually are near to a known constant value. EXAMPLE GEOMETRY REVISITED We return to the example geometry used previously. It has 31 total satellites above 5° including two geostationary satellites. For the purposes of this ARAIM analysis we will discard these two geostationary satellites and only evaluate the remaining 29 satellites. We have further assumed that the probability of satellite failure, Psat, is 10-5, the probability of constellation failure, Pconst, is 10-4, the integrity confidence bound, URA/SISA, is 1 m, the accuracy bound, URE/SISE, is 0.67 m, and the nominal bias bound, bnom, is 0.75 m. We have assumed that these values apply identically to each satellite.

The greedy selection method can also very effectively be applied to ARAIM [13]. Two ARAIM specific metrics were evaluated: the ARAIM VPL and the ARAIM vertical accuracy estimate, σv. The ARAIM VPL involves evaluation of numerous subsets, its specific formulation can be found in Annex A of [4]. The vertical accuracy estimate can be very similar to the square root of the SBAS vertical covariance term c3,3 (when the ratio of the ARAIM accuracy values is similar to the ratio of the SBAS confidence terms) and therefore minimizing this term is often similar to minimizing the SBAS VPL. Table 4 shows the results for four different selection algorithms: using the highest elevation angle satellites, the greedy algorithm selecting the best ARAIM VPL at each step, the greedy algorithm selecting the best vertical accuracy estimate at each step, and an optimal method that selects the smallest VPL over all possible combinations. Unlike for SBAS, ARAIM HPLs and VPLs can improve by removing satellites. This is because the least squares weights used for ARAIM do not necessarily minimize the VPL, which includes bias terms. Here it is obvious that the last three methods all perform much better at limiting VPL growth. In this example, all subset geometries have VPLs that are slightly below the all-in-view case. This situation is not uncommon when there are many satellites and the protection levels are small. The SBAS protection levels in Table 1 are all smaller than the corresponding values in Table 4. This is to be expected deep into SBAS coverage where nearly all satellites have a good SBAS correction. However, on the edge of coverage, only some satellites will be corrected by SBAS, but all will likely be usable by ARAIM. A possible algorithm would be to compare the all-in-view SBAS and ARAIM protection levels and then optimize for whichever one has better performance. In regions of good SBAS coverage, SBAS would be preferred. At the edges and out of SBAS coverage ARAIM would be preferred. OPERATIONAL CONSIDERATIONS An issue not addressed in this paper is the timing of making and changing selections. When choosing the best satellites to track, it is important to remember that it can take a little while to lock onto a satellite and establish tracking. If the satellite has not been observed recently, the receiver will need to obtain the broadcast ephemeris and confirm it with a second decoding. Thus it can take over a minute from deciding to track a satellite before it can be used in the position solution. Thus, one should not attempt to change their selected set of satellites too often.

Table 4. ARAIM VPL and HPL for differing numbers of channels and the different selection methods.

Table 4. ARAIM VPL and HPL for differing numbers of channels and the different selection methods.

Elevation Greedy VPL Greedy σv Optimal VPL Channels HPL VPL HPL VPL HPL VPL HPL VPL

29 8.59 12.11 8.59 12.11 8.59 12.11 8.59 12.11 28 8.58 12.49 8.48 11.70 8.89 11.89 8.48 11.70 27 8.61 12.26 8.69 11.61 8.83 11.87 8.38 11.59 26 8.74 12.13 8.63 11.55 9.39 11.83 8.84 11.51 25 8.73 12.88 8.54 11.49 10.81 11.78 8.91 11.47 24 8.67 12.62 8.91 11.45 11.2 11.76 8.46 11.42 23 11.11 17.87 8.87 11.41 11.13 11.74 8.59 11.38 22 10.99 17.29 8.83 11.38 11.03 11.72 8.85 11.34 21 11.00 17.25 9.11 11.34 10.96 11.7 9.69 11.30 20 11.18 17.83 9.55 11.31 12.22 11.66 10.62 11.27

Elevation Greedy VPL Greedy σv Optimal VPL Channels HPL VPL HPL VPL HPL VPL HPL VPL

29 8.59 12.11 8.59 12.11 8.59 12.11 8.59 12.11 28 8.58 12.49 8.48 11.70 8.89 11.89 8.48 11.70 27 8.61 12.26 8.69 11.61 8.83 11.87 8.38 11.59 26 8.74 12.13 8.63 11.55 9.39 11.83 8.84 11.51 25 8.73 12.88 8.54 11.49 10.81 11.78 8.91 11.47 24 8.67 12.62 8.91 11.45 11.2 11.76 8.46 11.42 23 11.11 17.87 8.87 11.41 11.13 11.74 8.59 11.38 22 10.99 17.29 8.83 11.38 11.03 11.72 8.85 11.34 21 11.00 17.25 9.11 11.34 10.96 11.7 9.69 11.30 20 11.18 17.83 9.55 11.31 12.22 11.66 10.62 11.27

Some priority may be given to satellites that are already being tracked. One may also want to be cautious about selecting too many low elevation satellites. These satellites typically have lower received power and are more susceptible to unexpected loss. Some low elevation satellites will also be in the process of setting and it may be preferable to select a replacement before the satellite goes below the elevation mask. Having a large number of channels and a large number of satellites in view will hopefully provide sufficient margin such that the loss of any one satellite will not result in a loss of service. There are many aspects to consider when considering the time evolution of satellite selection. However these are beyond the scope of this paper. CONCLUSIONS We have identified a weakness of the traditional elevation angle based selection algorithm when combined with a limited number of tracking channels. It has the potential to perform worse when more satellites are in view of the user. The VDOP and VPL become worse when low elevation satellites are removed in favor of higher ones. We have quantified this potential impact for an assumed set of different geometries. Nearly 50% increases in VPL and HPL are possible when assuming 24 channels as compared to the all-in-view solution that contained up to 31 satellites. We also presented an algorithm that does a much better job of selecting the satellites to track. This downdate algorithm limited the VPL growth to below 2% when considering 24 tracking channels under the same set of geometries. Further this algorithm is very efficient and does not require repeated evaluation of subset geometries. It acts on the all-in-view geometry to create a ranked list of which satellites are most important to track. Finally we propose a new specification method to evaluate performance rather than simply state a minimum number of tracking channels. The better the selection algorithm the fewer required tracking channels. Manufactures would also have the option to use a simpler algorithm, but at the cost of having a larger number of tracking channels.

ACKNOWLEDGMENTS The authors would like to gratefully acknowledge the FAA Satellite Product Team for supporting this work under MOA contract number DTFAWA-15-A-80019. The opinions expressed in this paper are the authors’ and this paper does not represent a government position on the future development of WAAS. REFERENCES [1] Minimum Operational Performance Standards for Global Positioning System/Wide Area Augmentation System Airborne Equipment, RTCA DO229D Change 1, Feb. 2013. [2] Blanch, J., Walter, T., Enge, P., Wallner, S., Fernandez, F., Dellago, R., Ioannides, R., Pervan, B., Hernandez, I., Belabbas, B., Spletter, A., and Rippl, M., “Critical Elements for Multi-Constellation Advanced RAIM for Vertical Guidance,” NAVIGATION, Vol. 60, No. 1, Spring 2013, pp. 53-69. [3] Milestone 2 Report, EU-U.S. Cooperation on Satellite Navigation, Working Group-C ARAIM Technical Subgroup, February 11th, 2015. available at http://www.gps.gov/policy/cooperation/europe/2015/ working-group-c/ARAIM-milestone-2-report.pdf [4] Milestone 3 Report, EU-U.S. Cooperation on Satellite Navigation, Working Group-C ARAIM Technical Subgroup, February 26th, 2016. available at http://www.gps.gov/policy/cooperation/europe/2016/ working-group-c/ARAIM-milestone-3-report.pdf [5] Phatak, M. S., “Recursive method for optimum GPS satellite selection,” in IEEE Transactions on Aerospace and Electronic Systems, vol. 37, no. 2, pp. 751-754, Apr 2001. doi: 10.1109/7.937488 [6] Kihara, M. and Okada, T., “A Satellite Selection Method And Accuracy For The Global Positioning System,” NAVIGATION, Journal of The Institute of Navigation, Vol. 31, No. 1, Spring 1984, pp. 8-20. [7] Blanco-Delgado, N. and Nunes, F.D., “A Convex Geometry Approach to Dynamic GNSS Satellite Selection for a Multi-Constellation System,” Proceedings of the 22nd International Technical Meeting of The Satellite Division of the Institute of Navigation (ION GNSS 2009), Savannah, GA, September 2009, pp. 1351-1360.

[8] Miaoyan, Z., Jun, Z., Yong, Q., “Satellite Selection for Multi-Constellation,” Proceedings of IEEE/ION PLANS 2008, Monterey, CA, May 2008, pp. 1053-1059. [9] International Standards and Recommended Practices (SARPS), Annex 10 – Aeronautical Telecommunications, Vol. 1I – Radio Navigation Aids, 6th edn., ICAO, July 2006. [10] Gerbeth, D., Felux, M., Circiu, M., and Caamano, M., “Optimized Selection of Satellite Subsets for a Multi-constellation GBAS,” Proceedings of the 2016 International Technical Meeting of The Institute of Navigation, Monterey, California, January 2016, pp. 360-367. [11] Jan, S.S., Chan, W. Walter, T., and Enge, P., “Matlab Simulation Toolset for SBAS Availability Analysis,” Proceedings of the 14th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GPS 2001), Salt Lake City, UT, September 2001, pp. 2366-2375. [12] Walter, T. and Blanch, J., “Characterization of GNSS Clock and Ephemeris Errors to Support ARAIM,” Proceedings of the ION 2015 Pacific PNT Meeting, Honolulu, Hawaii, April 2015, pp. 920-931. [13] Kropp, V., “Reduced All-in-View Satellite Set in ARAIM user algorithm,” in preparation for publication.